How to change the camera from Landscape to Portrait
Probably the easiest way to do this is to swap the values for the x and y resolution fields of the camera Dimensions-Resolution section. So, for instance, change it from 1920x1080 to 1080x1920.



For 2.8 the resolution settings are in Properties > Output > Resolution



The other way I know of is to select the camera and, while viewing from it, rotating the camera 90 degrees. This, however, rotates the view too.